Reconstruction of Binary Shapes From Blurred Images via Hankel-Structured Low-Rank Matrix Recovery

No Relaxation: Guaranteed Recovery of Finite-Valued Signals from Undersampled Measurements

“…Theorem 1 shows that it is possible to downsample the output of the filter h by a factor of L and yet uniquely identify a finite-valued input to the filter. In contrast to existing approaches [13,12,7], this show that the finite valued constraint alone ensures injectivity of the overall map and it is not necessary to impose any additional sparsity constraint.…”

“…A special case of this model involves binary valued signals (i.e. D " 1) and such binary valued signals, shapes or images have been considered in [7,12,13]. However, they relax the binary constraint to recover x using convex optimization, and theoretical guarantees are limited to random sampling.…”

“…The recovery of finite-valued signal x from undersampled measurements is potentially an ill posed problem when M ăă N . Most existing approaches utilize sparsity constraints and tools from compressed sensing to solve this problem [11,12,7,5,13]. However, these approaches end up relaxing the finite-valued (binary) constraint to develop convex relaxations of the original non-convex problem.…”

“…Secondly, we develop a novel decoding algorithm (inspired by the idea of β´expansion) that can simultaneously enforce the finite-valued constraint and perform computationally efficient decoding from subsampled measurements. Unlike prior works [7,11,13], our theoretical guarantees are applicable for uniform subsampling case and do not require any additional sparsity constraint.…”

“…Recently, the reconstruction of binary shapes and images from their convolution with a finite length filter was considered in [7,8], where the authors utilize the sparse structure of the image to formulate a low-rank matrix recovery problem to obtain the desired image. Another instance of such finitevalue constraint occurs in the recovery of piece-wise constant images [1,9,10].…”